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Performance of subband HFB-based A/D converters
Jacques Oksman, Davud Asemani
To cite this version:
Jacques Oksman, Davud Asemani. Performance of subband HFB-based A/D converters. International
Symposium on Signal Processing and its Applications, Feb 2007, Sharjah, United Arab Emirates.
pp.CD-ROM Proceedings. �hal-00261164�
Performance of subband HFB-based A/D converters
Davud Asemani, Jacques Oksman
Department of Signal Processing and Electronic Systems Ecole Sup´erieure d’Electricit´e (Supelec)
91192, Gif sur Yvette, France Email: [email protected]
Abstract— Subband Hybrid Filter Bank (HFB) structures may replace the classic ones in parallel A/D Converters (ADC). In this paper, subband and original HFB-based A/D converters are simulated and compared in the time domain. For this purpose, firstly a brief survey on subband HFB-based A/D converters is presented. According to the simulations, subband HFB-based ADC exhibits a lower sensitivity to realization errors than classic one. Besides, compensation techniques such as noise canceling may be used for correcting the analog imperfections of subband HFB. This is not possible for classic HFB. The output resolution of subband HFB-based ADC is shown to be better than the one related to classic architecture using FIR synthesis filters with the same order for some exemplary inputs. Computation complexity related to subband architecture is compared with the one due to classic structure. It may be seen that subband HFB-based ADC includes less computational complexity than the classic architecture if an adaptive compensation of analog imperfections is considered.
I. INTRODUCTION
The parallel structures of A/D converters may be applicable for wide-band analog signals [1]. The Hybrid Filter Bank (HFB) structure of A/D converters has been proposed as a candidate for realizing parallel A/D converters [2]. The original HFB-based ADC is shown in figure 1 [3]. Aliasing
MT
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ADC
ADC
ADC M
M M Analysis Part
(MT) (T)
n n
H0(s)
H1(s)
x0[n]
x1[n]
HM−1
xM−1[n]
F0(z)
F1(z)
y[n]
FM−1 x(t)
t
Fig. 1. The original architecture of M-channel maximally-decimated A/D converter using HFB structure. It consists of the analysis (analog) and synthesis (digital) filter banks.
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ADC
ˆs0[n]
ˆs1[n]
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n (MT)
x(t)
t
H0(s)
H1(s)
x0[n]
x1[n] F(z)
Synthesis Filters Matrix
xM−1[n]
HM−1(s)
Fig. 2. The subband architecture of HFB-based A/D converter for estimating the subband components of input signal. The outputs ˆs0[n], ˆs1[n], ..., and ˆsM−1[n] are the estimated subband signals.
terms are interferences that restrict the resolution of classic HFB-based A/D converters [4]. HFB-based A/D converters may exhibit an acceptable resolution using FIR synthesis filters if a small oversampling ratio is used [5]. However, they are very sensitive to realization errors so that the output resolution reduces drastically when the analysis filters of HFB include even small errors [7]. Sensitivity to realization errors increases too when oversampling process is used [7]. Analog imperfections are unavoidable because of fabrication errors and ambient parameters such as temperature [6]. Subband HFB architecture has been recently proposed for A/D conversion as shown in figure 2 [8]. Based upon an LTI Multiple-Input Multiple-Output (MIMO) model of the analysis part, it uses a matrix of synthesis filters (Fig. 2). Subband HFB-based A/D converter behaves like a Frequency Division Multiplexing (FDM) receiver [8]. It assumes that the input signal is divided into M narrow-band signals. The decimated version of these signals are called subbands and are estimated as the outputs of the subband structure. The resolution of subband HFB- based ADC is limited by Inter-Channel Interference (ICI)
terms [8]. The aim of this paper is to compare the perfor- mance of both original and subband HFB structures in terms of interference and sensitivity to realization errors. For this purpose, both structures have been simulated in time-domain using Simulink/MATLAB. To better focus on the aliasing and ICI interferences, the quantization noise is neglected in this paper [4]. In the next section, the subband HFB-based A/D converter is reviewed and the design method of its synthesis filters matrix is also described. Both (original and subband) HFB structures are simulated in time-domain in section III and the residual interferences are exhibited. Section III provides also a comparison between two architectures in terms of computational complexity, output resolution and sensitivity to realization errors. The two structures are discussed in terms of the capability according to which analog imperfections may be compensated as well. At last, the conclusion is summarized in section IV.
II. SUBBANDHFB-BASEDA/DCONVERTER
Subband HFB structure has been proposed to provide a par- allel A/D converter using analog analysis and digital synthesis filters [8]. It provides M output signals for an M-channel structure (Fig. 2). The outputs ˆs0[n], ˆs1[n], ..., ˆsM −1[n] are the estimated forms of M input subbands s0[n], s1[n], ..., sM −1[n]. These outputs are associated with the sampling rate M T1 where T1 stands for the overall sampling rate. It is supposed that the wide-band input x(t) always respects the Nyquist criterion to avoid any spectral overlapping. Figure 3
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filters
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ˆs1[n]
ˆsM−1[n]
x0[n]
x1[n]
xM−1[n]
s1[n]
s0[n]
sM−1[n]
Synthesis matrix
Fig. 3. The discrete-time model of subband HFB-based A/D converter.
Analysis part has been modeled using virtual subband filtersGk(z).
exhibits schematically the model of subband HFB-based A/D converter in discrete-time domain when the quantization noise of ADCs is neglected [8]. The input subbands s0[n],..., and sM −1[n] may be virtually extracted from the original analog input x(t) (Fig. 3). Each subband filter Gk(ejω) (0 ≤ k ≤ M − 1) is a virtual filter with the following definition for
|ω| ≤ π [8]:
Gk(ejω) =
1 kMπ ≤ |ω| ≤ (k + 1)Mπ
0 else (1)
The analysis filter matrix H(z) includes M2 digital filters which are obtained from the original analog analysis filters [8].
For obtaining the elements of H(z), the intermediate digital filter Hk(ejω) is defined in terms of analog analysis filter Hk(jΩ) as following:
Hk(ejω) = 1 T
+∞
m=−∞
Hˇk(jΩ −2π T m)
Ω=Tω
(2)
where,
Hˇk(jΩ) =
Hk(jΩ) Ω ∈ [−πT, +Tπ]
0 elsewhere
Hk,r(ejω) is the (k, r)th element ofH(z). It is associated with thekthanalysis filterHk(jΩ). If r is even (r = 0, 2, ...), it is described as:
Hk,r(ejω) =
M1Hk(ejMω+jrMπ) ω ∈ [0, π]
M1Hk(ejMω−jrMπ) ω ∈ [−π, 0] (3) and in the odd case (r = 1, 3, ...), it is:
Hk,r(ejω) =
1
MHk(ejMω−j(r+1)Mπ) ω ∈ [0, π]
1
MHk(ejMω+j(r+1)Mπ) ω ∈ [−π, 0]
(4) Hk(ejω) is obtained from the analog analysis filter Hk(jΩ) ac- cording to (2). So, the analysis filter matrixH(z) is obtained.
To design the matrix of synthesis filters, Perfect Reconstruc- tion (PR) conditions are considered. IfF(ejω) represents the synthesis filters matrix, the PR equation will be:
F(ejω).H(ejω) = I.e−jωnd (5) where I is the M × M identity matrix and nd stands for an arbitrary delay. nd has been considered for maintaining the causality of synthesis filters. Using Least Squares (LS) optimization, the equation 5 leads to the following solution at each frequencyω:
F(ejω) = e−jωndHH(ejω)
H(ejω)HH(ejω)
−1 (6) where superscript (.)H stands for conjugate-transpose opera- tion. This relation may be established forN frequency points (N >> 1) [5]. Thus, the frequency response of each synthesis filter Fij(ejω) is obtained using (6). An FIR filter may be considered for estimating the (i, j)th element Fi,j(ejω) of the synthesis filters matrix. Using FIR estimation of synthesis filters, the equality (5) does not hold exactly. So:
T(ejω) = F(ejω)H(ejω) (7)
This exhibits that the kth output ˆsk[n] may be described in the frequency domain as:
Sˆk(ejω) = Tk,k(ejω)Sk(ejω) +
M −1
m=0,m=k
Tk,m(ejω)Sm(ejω) (8) where Tk,k(ejω) is the distortion function and Tk,m(ejω) is the ICI component related to the mth subband input.
ICI terms are similar to aliasing terms considered in the original HFB structure. The output resolution is restricted by ICI and aliasing terms for subband and classic architectures respectively.
III. MAIN RESULTS
In this section, the PR equation will be used for design of synthesis filters. To implement the HFB structures, a simply realizable analysis filter bank is considered and FIR synthesis filters are designed. Assuming an 8-channel HFB, the analysis filter bank includes an RC (first-order) circuit along with seven RLC (second-order) circuits. Using this set of analysis filters, the analysis filter matrix has been calculated according to (3) and (4). Then, synthesis filter bank is obtained using (6).
Each element Fi,j(ejω) of the synthesis filters matrix may be approximated by an FIR synthesis filter. In the subband case, some percents of each subband are considered as guard band to provide an acceptable performance [8]. In this paper, guard band ratio is supposed to be 10%. The optimum value of oversampling ratio (7%) is also used for the classic HFB case [7].
−0.5 0 0.5
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−120
−100
−80
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−40
Frequency response of error signal
dB
Normalized frequency
Oversampling area Original HFB
Subband HFB Guard band
Input sinusoid
Fig. 4. The FT of error signal (dB) versus normalized frequencies for original and subband HFB-based ADC. Analog input is a sinusoidal signal at the first subband. Each FIR synthesis filter has64 coefficients. There is a shift in the frequency of input sinusoid due to subband HFB because related output is decimated.
• Resolution
Both structures have been simulated in time-domain consid- ering different input signals. Considering a sinusoidal signal at the frequency 0.58Tπ (in the middle of first subband), the Fourier Transform (FT) of the error signal is shown in figure 4.
All outputs of subband structure are null except the output of the first one ˆs0[n]. It may be seen that there is only one
peak (at positive frequencies) in the error signal due to the subband architecture. It is associated with the input sinusoid frequency with a shift in the frequency domain because of decimation. However, the error signal due to the original structure includes eight (M = 8) error peaks (Fig. 4). In other words, it may be interpreted that the error related to one input subband appears at the other M − 1 subbands on the output of M-channel original HFB-based ADC. To better compare the output resolutions, a chirp signal is also considered as input. The frequency of this chirp is swept between 0.18Tπ and0.98T7π. The outputs of subband HFB-based ADC are null except for the first subband. Figure 5 illustrates the FT of
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−80
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Frequency response of error signal
dB
Normalized frequency Original HFB
Subband HFB Oversampling area
Guard band
Fig. 5. The FT of error signal (dB) versus normalized frequencies for original and subband HFB-based ADC considering a chirp as input signal.
The oversampling and guard band ratios are7% and 10% respectively.
the error signal versus the normalized frequencies for original and subband structures using FIR synthesis filters with 64 coefficients. It shows that post-filtering is essential for original HFB to suppress the oversampling band (here |f| ≥ 0.92T7π) even though there is no input component in oversampling area (Fig. 5). The output resolutions of the two structures are shown in table I for FIR synthesis filters with64 and 128 coefficients.
The resolution due to the original HFB is related to the case where post-filtering has been applied. It may be seen that subband HFB-based ADC provides a better resolution than the original structure supposing FIR synthesis filters with the same order.
TABLE I
RESOLUTION OFHFB-BASEDADCASSUMING A CHIRP SIGNAL AS INPUT(LREPRESENTS THE NUMBER OF COEFFICIENTS FORFIR
SYNTHESIS FILTERS).
Resolution (bits)
Structure L=64 L=128
Original HFB 9.5 11.2
Subband HFB 17.6 20
• Sensitivity analysis
The electronic elements are often associated with a nominal value plus a deviation or realization error. The subband and original structures have been simulated considering Gaussian- distributed realization errors for the electronic elements of analysis filter bank. The simulations were repeated for 1000 trials of random mutually-independent realization errors. Ana- log input signal consists of eight sinusoidal signals which cover the whole spectrum so that there is one sinusoid at the middle of each subband. Figure 6 shows the average resolution
0 2 4 6 8 10
0 2 4 6 8 10 12
Output resolution (bits)
STD of realization errors (%)
Bit Subband HFB
Original HFB
FIR synthesis filters with 64 coefficients
Fig. 6. The resolution (bits) of original and subband HFB-based ADC versus STD of realization errors. Gaussian errors have been applied to analysis filters.
FIR synthesis filter bank includes 64 coefficients.
of each HFB structure versus STandard Deviation (STD) of error distribution using64 coefficients for FIR synthesis filters.
It may be seen that the sensitivity of subband HFB-based A/D converter to realization errors is less than the one related to the classic architecture. Some correcting methods such as blind deconvolution and noise canceling may be used for digitally compensating the realization errors [9]. Those methods may be applied to the subband HFB because input and output are related through an LTI relation (Fig. 3). This is not valid for the original HFB since there is a time-varying operation of decimation between its input and output [8]. Compensation techniques may even provide a better resolution in the case of no realization error for subband architecture using the outputs of the other subbands.
• Computational complexity
The classic HFB-based A/D converter consists of M FIR synthesis filters, but the subband architecture needs M2 ones (see figures 1 and 2). For an FIR filter with L coefficients, L multiplying operations and delay components are effec- tively necessary. Then, for implementing the synthesis stage, subband architecture will need M2L multiplications to be compared withML ones in the classic case. Although, it does not require the upsampling operations (zero-padding by M).
To compare the computational complexity, one has also to consider the design phase from which FIR synthesis filters are designed. Assuming N frequency points for the design of the synthesis filters, original HFB structure would require the
inversion of an MN × MN matrix [5]. The subband HFB needs the inversion operation of N matrices of M × M [8].
In practice, N must be much larger than M (N >> M) to have an acceptable interpolation. Thus, the design phase of subband architecture is obviously much more complex than the classic one. The complexity of the design phase is particularly important when an adaptive method is applied to estimate the real analysis filter bank for compensating realization errors.
For example, the design phase of synthesis filters has to be regularly repeated to compensate the variations due to temperature. Accordingly, subband HFB-based ADC may be preferred to the original structure when realization errors are regularly compensated.
IV. CONCLUSION
In this paper, a brief survey on the subband HFB-based A/D converters was presented. The Perfect Reconstruction (PR) conditions related to this architecture were expressed, and ICI terms associated with each subband signal were formulated as well as the distortion function. The subband HFB structure of A/D converters has been compared with the original architec- ture. It was shown that subband HFB may provide a better resolution than original architecture for exemplary inputs of sinusoidal and chirp signals. Subband HFB also exhibits a lower sensitivity to realization errors than original HFB struc- ture. However, it is necessary to mention that subband HFB- based ADC may include a digital compensation stage using some techniques such as noise canceling. This is not applicable for original HFB-based A/D converters. Subband HFB-based A/D converters require M2 FIR filters in the synthesis stage but the original architecture uses onlyM FIR synthesis filters.
Nevertheless, the overall computational complexity of subband HFB-based A/D converters is lower than the one related to original architecture if a regular compensation of realization errors is considered.
REFERENCES
[1] Brown J.L., ”Generalized sampling and the reconstruction problem for maximally decimated filter banks”, IEEE Int. Conf. on Acoustics, Speech and Signal Processing (ICASSP), 1989, pp.1195-1198.
[2] Oliaei O., ”High-Speed A/D and D/A converters using Hybrid Filter Banks”, IEEE International Conference on Electronics, circuits and Systems (ICECS), Portugal, Sept., 1998, pp.143-146.
[3] Velazques S.R., ”Hybrid Filter Banks for Analog/Digital Conversion”, Phd Thesis, Massachusetts Institute of Technology, June 1997.
[4] Lowenborg P., Johansson H., Wanhammar L., ”Two-channel digital and hybrid Analog/Digital multirate filter banks with very low-complexity analysis or synthesis filters”, IEEE Transactions on circuits and Systems, Vol.50, No.7, July, 2003, pp.355-367.
[5] Petrescu T.G., Oksman J., ”Synthesis of hybrid filter banks for A/D con- version with implementation constraints: optimized frequency response approach” , 46thIEEE Midwest Symposium on Circuits and Systems (MWSCAS), Cairo, Egypt, Dec., 2003.
[6] Pinheiro M. et al., ”Improving the near-perfect HFB performance in the presence of realization errors”, ICASSP’01, 2001, vol. 2, pp.1069-1072.
[7] Asemani D., Oksman J., ”Influences of oversampling and analog imper- fections on hybrid filter bank A/D converters”, MWSCAS’06, San Juan, Puerto Rico, Aug., 2006.
[8] Asemani D., Oksman J., ”Subband architecture for HFB-based A/D con- verters ”, International symposium on communications and information technologies, Bangkok, Thailand, Oct., 2006.
[9] Haykin Simon (Editor), ”Blind Deconvolution”, Prentice Hall, 1994.
